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Guangcun Lu

Publications and source records attributed to Guangcun Lu.

At least 19 recordsLinked to original sources
arXiv2026

Bifurcations of periodic and antiperiodic orbits near an equilibrium in autonomous differential delay systems with one or two delays

We investigate bifurcations of periodic and antiperiodic solutions near equilibria for four classes of potential differential-delay equations involving one or two delays. By reformulating each system as a (generalized) Hamiltonian system, and applying the Hamiltonian bifurcation theory recently developed by the author we obtain bifurcation results of both Fadell--Rabinowitz and Rabinowitz type. The analysis relies on the computation of Maslov--type indices for fundamental solutions of the associated linear Hamiltonian systems.

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arXiv2026

Bifurcation of periodic and antiperiodic solutions in non-autonomous potential-type delay systems

As a continuation of our previous work, where bifurcations of periodic and antiperiodic orbits near an equilibrium were studied for autonomous differential delay systems with one or two delays, this paper investigates bifurcations of periodic and antiperiodic solutions for five classes of non-autonomous potential delay differential systems with multiple delays. By reformulating these systems as generalized Hamiltonian systems on Euclidean spaces, we apply the Hamiltonian bifurcation theory recently developed by the author to establish alternative bifurcation results of both Fadell--Rabinowitz and Rabinowitz type.

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arXiv2026

Bifurcations for Lagrangian systems and geodesics I

This paper is Part I of a two-part series. We investigate bifurcation phenomena in Lagrangian systems with various boundary conditions and constraints, focusing on the interplay between Morse theory and the existence of multiple solutions through three principal configurations: Lagrangian trajectories connecting two submanifolds or with endpoints related by an isometry, and brake orbits in Lagrangian systems. For each configuration, we establish necessary and sufficient conditions for bifurcation using Morse index and nullity techniques, including classification of Rabinowitz-type alternative bifurcation scenarios. For Euler-Lagrange curves emanating perpendicularly from a submanifold, we develop a unified Morse-theoretic framework that rigorously connects geometric focal structure (e.g., conjugate points) and analytic bifurcation behavior (e.g., solution branching patterns).

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arXiv2024

Bifurcations for Lagrangian systems and geodesics II

This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in earlier work and the results of Part I, this study makes contributions in two main directions. In Part A, we focus on bifurcations of generalized periodic solutions in autonomous Lagrangian systems. By employing Morse index and nullity techniques within the normal space to the $\mathbb{R}$-orbits of solutions, we derive necessary and sufficient conditions for bifurcation, encompassing scenarios of both Fadell--Rabinowitz and Rabinowitz type. In Part B, we extend these results to the geometric setting of geodesic bifurcations in Finsler and Riemannian manifolds. A principal achievement is the significant refinement of the classical Gauss lemma and its generalizations by Morse-Littauer and Savage, providing a precise description of geodesic behavior near critical points of the exponential map. The sharpness of these theoretical results is rigorously tested and confirmed through explicit counterexamples, such as the round sphere. The work is technically rigorous, leveraging a specialized technique developed by the author to establish novel bifurcation theorems. These findings have profound theoretical implications and potential applications in related fields such as the Zermelo navigation problem and the study of stationary spacetimes.

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arXiv2023

Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

In this paper we construct analogues of Ekeland-Hofer and Hofer-Zehnder symplectic capacities based on a class of Hamiltonian boundary value problems motivated by Clarke's and Ekeland's work, and study generalizations of some important results about the original two capacities (for example, the famous Weinstein conjecture, representation formula for $c_{\rm EH}$ and $c_{\rm HZ}$, and a theorem by Evgeni Neduv).

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arXiv2023

Coisotropic Hofer-Zehnder capacities of convex domains and related results

We prove representation formulas for the coisotropic Hofer-Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by Lisi and Rieser recently), study their estimates and relations with the Hofer-Zehnder capacity,give some interesting corollaries, and also obtain corresponding versions of a Brunn-Minkowski type inequality by Artstein-Avidan and Ostrover and a theorem by Evgeni Neduv.

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arXiv2023

Coisotropic Ekeland-Hofer capacities

For subsets in the standard symplectic space $(\mathbb{R}^{2n},ω_0)$ whose closures are intersecting with coisotropic subspace $\mathbb{R}^{n,k}$ we construct relative versions of the Ekeland-Hofer capacities of the subsets with respect to $\mathbb{R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb{R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal{S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal{S}$.

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arXiv2023

A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012.

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arXiv2022

The Poisson bracket invariant for open covers consisting of topological disks on surfaces

L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension $2$. In this note, instead of open cover consisting of displaceable sets in their work, we consider open cover constituted of topological discs and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.

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arXiv2021

Bifurcations for Hamiltonian systems

With the dual variational principle and the saddle point reduction we use the abstract bifurcation theory recently developed by author in previous work to prove many new bifurcation results for solutions of four types of Hamiltonian boundary value problems nonlinearly depending on parameters. The most interesting and important among them are those alternative results which can only be proved with our generalized versions of the famous Rabinowitz's alternative bifurcation theorem.

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arXiv2021

Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.

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arXiv2021

Combinatorial formulas for some generalized Ekeland-Hofer-Zehnder capacities of convex polytopes

Motivated by Pazit Haim-Kislev's combinatorial formula for the Ekeland-Hofer-Zehnder capacities of convex polytopes, we give corresponding formulas for $Ψ$-Ekeland-Hofer-Zehnder and coisotropic Ekeland-Hofer-Zehnder capacities of convex polytopes introduced by the second named author and others recently. Contrary to Pazit Haim-Kislev's subadditivity result for the Ekeland-Hofer-Zehnder capacities of convex domains, we show that the coisotropic Hofer-Zehnder capacities satisfy the subadditivity for suitable hyperplane cuts of two-dimensional convex domains in the reverse direction.

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arXiv2021

Higher $P$-symmetric Ekeland-Hofer capacities

This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for $P$-symmetric subsets in the standard symplectic space $(\mathbb{R}^{2n},ω_0)$, which is motivated by Long and Dong's study $P$-symmetric closed characteristics on $P$-symmetric convex bodies. We study the relationship between these capacities and other capacities, and give some computation examples. Moreover, we also define higher real symmetric Ekeland-Hofer capacities as a complement of Jin and the second named author's recent study of the real symmetric analogue about the first Ekeland-Hofer capacity.

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arXiv2020

A note on symmetrical symplectic capacities

For a convex domain in the standard Euclidean symplectic space which is invariant under a linear anti-symplectic involution $τ$ we show that its Ekeland-Hofer-Zehnder capacity is equal to the $τ$-symmetrical symplectic capacity of it.

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